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5 MATRICES 371
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pA are linearly independent over K'. Hence the p^ba are linearly independent
over K', and our assertion is proved. In particular: |
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(A.4.21) If E is finite dimensional over K, and K is finite dimensional over K',
then E is finite dimensional over K' and we have
(A.4.21.1) dimK, E = dimK E • dimK, K.
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5. MATRICES
(A.5.1) Let E, F be vector spaces over a field K, and suppose that E is of
finite dimension n. Let (at-)1^I-^n be a basis of E, so that E is the direct sum of the Kat. By (A.3.1) there is a one-one correspondence between the linear mappings u of E into F and the families (b^^t^ of n vectors of F; this correspondence is defined by bt = u(at) (1 ^ i < n). Thus the vector space Hom(E, F) is isomorphic to Fn. |
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(A.5.2) Suppose furthermore that F is of finite dimension m, and let
(bj)i^jt$m be a basis of F. Then there is a one-one correspondence between the vectors y e F and the families (^j)i^j^m °f m elements of K, defined by |
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y= £*?/&/. Hence there is a one-one correspondence between the linear
j«i
mappings u : E -* F and the " double " families (o^) (with 1 <y < w, 1 < / < «)
of elements of K, defined by the relations
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Such families are called matrices with m rows and n columns (or m x n
matrices) over K. They form a vector space over K, isomorphic to Kmn. The subfamily (o,,)^^ is theyth row, and the subfamily (a/Di*^™ is the zth column, of the matrix (ajj)1<</iSWfl<j<B. The matrix M(u) == (o^) defined by (A.5.2.1) is called the matrix ofu with respect to the bases (at) and (&,-). If w, u' are two linear mappings of E into F, and if 1 is any scalar, then
M(u + iO = M(u) + M(u'\
M(lu) = AAf («),
the matrices being taken in each case with respect to the same bases (at) of E
and (bj) of F. |
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